The complete classification of unital graph $C^*$-algebras: Geometric and strong

Abstract

We provide a complete classification of the class of unital graph C * -algebras -prominently containing the full family of Cuntz-Krieger algebras -showing that Morita equivalence in this case is determined by ordered, filtered K-theory. The classification result is geometric in the sense that it establishes that any Morita equivalence between C * (E) and C * (F ) in this class can be realized by a sequence of moves leading from E to F , in a way resembling the role of Reidemeister moves on knots. As a key ingredient, we introduce a new class of such moves, establish that they leave the graph algebras invariant, and prove that after this augmentation, the list of moves becomes complete in the sense described above.Along the way, we prove that every ordered, reduced filtered K-theory isomorphism can be lifted to an isomorphism between the stabilized C * -algebras -and, as a consequence, that every ordered, reduced filtered K-theory isomorphism preserving the class of the unit comes from a * -isomorphism between the unital graph C * -algebras themselves.It follows that the question of Morita equivalence amongst unital graph C * -algebras is a decidable one. As immediate examples of applications of our results we revisit the classification problem for quantum lens spaces and verify, in the unital case, the Abrams-Tomforde conjectures.

On The Classification of Quantum Lens Spaces of Dimension at most 7

“…Since L r;m 2n+1 contains no sinks and two quantum lens spaces are stably isomorphic if and only if they are isomorphic by [7,Proposition 14.8], the above theorem boils down to:…”

Decidability of flow equivalence and isomorphism problems for graph C*-algebras and quiver representations

“…We now consider poset blocked matrices with a rectangular structure. This is natural, and necessary for showing that the work of [ERRS16] implies general decidability results for unital graph C * -algebras. The adjacency matrices for these (directed) graphs have only finitely many vertices, but may have infinitely many edges; the analysis of their adjacency matrices (with "∞" an allowed entry) is reduced in [ERRS16] to the analysis of associated rectangular matrices with integer entries.…”

“…We impose the nontriviality requirement that I and J are nonempty, with I ∪ J = P. For R a subring of C, define M P, m, n (R) to be the set of m × n matrices with I × J block form, with M ij an m i × n j matrix over R such that M i,j = 0 implies i j. (As in [ERRS16], m i = 0 can be viewed as producing an empty block row indexed by i, and similarly n j = 0 corresponds to an empty block column.) I and J are posets, with the order inherited from P.…”

“…The interaction in the case of general graph C * -algebras is less direct but still signficant. For much more on these algebras and their classification, we refer to the discussion and references of [ERRS16]. Now, let (P, ) be a finite poset.…”

“…In related work, Boyle and Huang reduced the question of deciding flow equivalence for shifts of finite type to deciding whether two matrices A, B ∈ M P, n (Z) are SL P, n (Z)equivalent (see [Boy02]; Huang's alternate development was never published). Eilers, Restorff, Ruiz and Sørensen [ERRS16] reduced the decidability of stable isomorphism of unital graph C * -algebras (a class including the Cuntz-Krieger C * -algebras) to a more general problem of poset blocked equivalence of rectangular matrices, which we describe later. They also reduced decidability of isomorphism of unital graph C * -algebras to yet another matrix equivalence problem.…”